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This is a pretty great question with a fairly subtle answer. It is not much more than just flipping a coin, indeed!

See section "4. Relativistic Causality" of http://www.scottaaronson.com/democritus/lec11.html for the best explanation I know of. This entire book "Quantum Computing Since Democritus" is absolutely great if you want to understand these topics.

See also https://en.wikipedia.org/wiki/No-communication_theorem


Think of it this way: We've got two players, Alice and Bob, and they're playing the following game. Alice flips a fair coin; then, based on the result, she can either raise her hand or not. Bob flips another fair coin; then, based on the result, he can either raise his hand or not. What both players want is that exactly one of them should raise their hand, if and only if both coins landed heads. If that condition is satisfied then they win the game; if it isn't then they lose.

They can win 75% of the time if they just never raise their hands. Using a shared entangle state they can "cheat" and win 85.3% of the time by using a specific protocol, because they rely on some new form of correlation. But they still can not use this correlation to send messages (see the no communication theorem). Naively (this naive intuition does break!), you can imagine them having two slightly correlated coins - sure, after Alice flips hers she knows Bob's result, but she did not decide the result of her coin so she can not use it to send information to Bob.

But here you have two independent coins. My question is, how is entanglement fundamentally different than having two sides of the SAME coin but not looking until later?

Let's forget how this looks like two correlated coins and focus on your question.

Yes, if Alice and Bob just measure the spin of their corresponding electron, indeed it just looks like half of a precut coin. But if they want to win the game they will do something more complicated. (side note: To use the typical physicist terminology, this precut coin is a hidden variable - something set inside of the electron even before its measurement.)

But Alice and Bob can do something more interesting, something that permits them to win the game we described more often than 75% of the time. This involves Alice measuring the spin of the electron in a projection different than the one in which Bob is measuring (this is part of the "specific protocol" that I mentioned in the previous post). So while Alice is checking whether the result is h=Heads or t=Tails, Bob is checking whether the result is h+t or h-t. Now the two electrons seize to behave like the two parts of the same coin (and this is what permits them to win the game more often than 75% of the time). Explaining how exactly they use this protocol would take too much space, but you can look it up in the link to the book I provided.

P.S. However, as explained in the previous post, this does not permit them to send messages to each other.

P.P.S. Again, Section 4 of the linked page explains this in details!

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